Going one step further as explained in [1]}, we can take the topological realizations of the above real cone stacks obtaining a morphism of topological stacks (i.e. geometric stacks over the category \(\operatorname{\bf {Top}}\)  of topological spaces):
\(\big \vert \Phi ^{trop}\big \vert \colon \big \vert \mathcal {J}^{trop}_{g,n}\big \vert \longrightarrow \big \vert \mathcal {M}^{trop}_{g,n}\big \vert .\) 
